In the heterogeneous network (HetNet) employing downlink non-orthogonal multiple access (NOMA), we focus on the non-convex optimization problem to optimize the spectral efficiency (SE) while the users satisfy the quality-of-service (QoS) requirement. In the previous work, the optimal joint successive interference cancellation and power allocation (JSPA) algorithm for maximizing SE is proposed to solve the mixed-integer non-linear programming (MINLP) problem in NOMA-enabled HetNet. However, the optimal solution requires exponential complexity by the number of base stations (BSs). Therefore, we present a deep neural network (DNN)-based algorithm for JSPA to reduce the complexity. In particular, to deal with the MINLP-based JSPA problem, we reformulate it into an equivalently simple problem that optimizes only the power consumption of BSs. Then, we introduce the unsupervised DNN-based method for JSPA to handle the simplified problem. The presented scheme yields improved SE and outage performance compared with traditional DNN-based methods. Additionally, we propose a user selection scheme with low complexity to enhance the SE of the proposed DNN-based power allocation. Through simulations, we illustrate that the suggested DNN-based scheme can attain SE performance similar to that of the optimal scheme.